Apollonius <Pergaeus> - Apollonii Pergaei Conicorvm Lib. V. VI. VII. paraphraste Abalphato Asphahanensi : nunc primum editi ; additvs in calce Archimedis assvmptorvm liber, ex codibvs arabicis mss Abrahamus Ecchellensis Maronita latinos reddidit, Jo. Alfonsvs Borellvs curam in geometricis versione contulit & [et] notas vberiores in vniuersum opus adiecit

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252214Apollonij Pergæi& parallelæ erunt alicui recta lineæ ex L

291[Figure 291] diuidenti angulum H L M, eo quod paral-
lelæ erãt rectæ H g diuidenti angulum L H
G, & prius B E vlterius, quàm C F ten-
debat ad partes asymptoti H I; ergo è con-
tra C F vlterius tendet ad partes asymptoti
H G, & educũtur ab asymptoto L M producta,
& parallelæ ſunt rectæ lineæ ex L diuidenti
angulũ H L M, contentum à recta linea cen-
tra coniungente, & a symptoto M L, in qua
illæ cadunt; igitur ( ex prima parte huius propoſitionis) C F maior erit, quàm
B E; & è contra B E vlterius tendens ad partes asymptoti H I minor erit, quã
C F; vt propoſitum fuerat.

Sint duæ æquales parabolæ A B, D E ad eaſdem partes cauæ, qua-
11PROP. 4.
Addit. rum diametri G I, H K ſint congruentes aut parallelæ inter ſe, nec nõ
ad eas ordinatim applicatæ B Z K, L X N ſint parallelæ alicui rectæ
diuidenti angulum G H K à recta linea G H vertices coniungenti, &
diametro H K interioris ſectionis D H contentum, ſi diametri congruentes
non fuerint. Dico quod, B E, L M portiones applicatarum à ſectioni-
bus ad eaſdem partes interceptæ, ſemper magis diminuentur, quo magis
à verticibus recedunt; efficienturque minores quacumque recta linea pro-
poſita, ſi diametri ſunt congruentes: ſi verò ſunt parallelæ nunquam mi-
nores erunt portione ordinatæ inter diametros intercepta. At ſi paral-
lelæ fuerint alicui rectæ lineæ diuidenti angulum H G I à recta G H,
& diametro I G exterioris ſectionis A G contentum, ſemper magis au-
gentur, ſed erunt ſemper minores ea quæ à diametris intercipitur. Vel ſi
fuerint parallelæ diametris non congruentibus, ſemper magis augentur,
quo magis à concurſu recedunt.

Sit F G latus rectum diametri G I in,

292[Figure 292] parabola G B, ordinatim applicatæ B E K,
& L M N ſecent diametrum G I in X, Z,
& diametrum H K in N, K, & ſecetur
abſciſſa G I æqualis H K, & G R æqualis
H N; ideoque R I æqualis erit N K, ſeu
X Z (propterea quod in parallelogrammo
N Z oppoſita latera æqualia ſunt) ducan-
turque ordinatæ O I, Q R, quæ erunt æqua-
22ex 10.
ex 21.
huius. les, & congruentes ipſis E K, M N pro-
pter æqualitatem ſectionum, & abſciſſarũ
ſimilium diametrorum; ducanturque à pun-
ctis E, L, Q rectæ lineæ E S, L T, Q V
parallelæ diametris occurrentes ipſis B E,
& O I in S, T, V: manifeſtum eſt S

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